351.
HYDEODYNAMICAL NOTES. [Philosophical Magazine, Vol. XXL pp. 177—195, 1911.]
Potential and Kinetic Energies of Wave Motion.—Waves moving into Water.—Concentrated Initial Disturbance with inclusion of Capillarity.-—Porio< in Deep "Water advancing without change of Type.—Tide Race.s.—Rotational Flu in a Corner.—Steady Motion in a Corner of Viscous Fluid.
IN the problems here considered the fluid is regarded as inconi] and the motion is supposed to take place in two dimensions.
Potential and Kinetic Energies of Wave Motion.
When there is no dispersion, the energy of a progressive, wav form is half potential and half kinetic. Thus in the case of a long shallow water, "if we suppose that initially the surface is displaced, the particles have no velocity, we shall evidently obtain (an in th sound) two equal waves travelling in opposite directions, who.se tofcal are equal, and together make up the potential energy of the orif. placement. Now the elevation of the derived waves must be half < the original displacement, and accordingly the potential energies k; ratio of 4:1. Since therefore the potential energy of each derive one quarter, and the total energy one half that of the original (lisp it follows that in the derived wave the potential and kinetic orir equal" *.
The assumption that the displacement in each derived wa separated, is similar to the original displacement fails when the n: dispersive. The equality of the two kinds of energy in an infi gressive train of simple waves may, however, be established as f
* "On Waves," Phil Mag. Vol. i. p. 257 (1876); Scientific Papers, Vol. I. p,. A correction is hen See Nicholson, Phil. Mag. Vol. xxv. p. 200, 1913.]emains to prove that z* necessarily exceeds n(n 4-1). That zli exceeds n2 is well known*, but this does not suffice. We can obtain what we require from a formula given, in Theory of Sound, 2nd ed. § 339. If the finite roots taken in order be zi} zz,... za..., we may write